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Theorem dfun2 3990
Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3991 and dfss4 3989 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation (class difference). (Contributed by NM, 10-Jun-2004.)
Assertion
Ref Expression
dfun2 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Proof of Theorem dfun2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3331 . . . . . . 7 𝑥 ∈ V
2 eldif 3713 . . . . . . 7 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 991 . . . . . 6 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
43anbi1i 733 . . . . 5 ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 eldif 3713 . . . . 5 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵))
6 ioran 512 . . . . 5 (¬ (𝑥𝐴𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 292 . . . 4 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
87con2bii 346 . . 3 ((𝑥𝐴𝑥𝐵) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
9 eldif 3713 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)))
101, 9mpbiran 991 . . 3 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
118, 10bitr4i 267 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)))
1211uneqri 3886 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382  wa 383   = wceq 1620  wcel 2127  Vcvv 3328  cdif 3700  cun 3701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-v 3330  df-dif 3706  df-un 3708
This theorem is referenced by:  dfun3  3996  dfin3  3997
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